= Positive-measure almost periodic indicator in a compact extension
{title2=$B=A\cap\varphi^{-1}C$}
Approximate $\mathbf1_A$ by relative almost periodic functions with summable squared global errors. <Tonelli theorem> makes their conditional errors tend to zero almost everywhere. Choose a positive base set on which the conditional mass of $A$ is bounded below, then use the <Egorov theorem> to make those errors uniform on a smaller positive set $C$. <Localization of relative almost periodicity to base sets> and conditional-norm closure prove $\mathbf1_B$ is relatively almost periodic and $\mu(B)>0$.
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